1. No category

Nuclear Reactions
in Lattice EFT
Gautam Rupak
Mississippi State University
Nuclear Lattice EFT Collaboration
Evgeny Epelbaum (Bochum)
Serder Elhatisari(Bonn)
Hermann Krebs (Bochum)
Timo Lähde (Jülich)
Dean Lee (NCSU)
Thomas Luu (Jülich)
Ulf-G. Meißner (Bonn)
Bethe Forum: Methods for Lattice Field Theory, Bonn, Germany
Mar 24, 2015
Motivation and Outline
• Reactions for Nuclear Astrophysics
• Adiabatic projection method
• neutron capture
• proton-proton fusion
• dimer-dimer scattering
Reactions in lattice EFT
• Consider: a(b, )c
• Need effective “cluster” Hamiltonian -- acts in
cluster coordinates, spins,etc.
• Calculate reaction with cluster Hamiltonian. Many
possibilities --- traditional methods, continuum
EFT, lattice method
Rupak & Lee, PRL 2013, arXiv:1302.4158
Pine, Lee, Rupak, EPJA 2013, arXiv:1309.2616
Rupak & Ravi, PLB 2014, arXiv:1411.2436
Adiabatic Projection
Method
~
Initial state |Ri
~ ⌧ =e
Evolved state |Ri
Microscopic Hamiltonian L3(A 1)
Cluster Hamiltonian L3
-- acts on the cluster CM and spins
⌧H
~
|Ri
Spin-1/2 Fermions
g(
†
"
" )(
†
#
#)
1.5x106
ann ⇠
19 fm, R ⇠ 1.4 fm
anp ⇠
24 fm, R ⇠ 1.4 fm
Potassium-40
Regal et. al, Nature 2003
a
1.0x106
5.0x105
0
scattering length (ao)
a
g⇠
m
r
]
[i@t +
2m
atom number
L=
†
2
2000
b
1000
0
-1000
-2000
-3000
215
220
225
B (G)
230
Neutron-Deuteron System
=
+
+···
=
+
+···
T (p) = h(p) +
Z
2⇡
1
T (p) =
µ p cot
dqK(p, q)T (q)
ip
Neutron-Deuteron System
−1
Convergence: L=7, b=1/100 MeV
25
E (MeV)
20
15
10
5
0
0
0.1
0.2
0.3
0.4
τ (MeV−1)
0.5
0.6
Pine, Lee, Rupak, EPJA 2013
Luscher’s Method
1
S(⌘), ⌘ =
p cot =
⇡L
✓
pL
2⇡
◆2
2
Efd
p
=
2µ⇤
B + ⌧ (p)[B
BL ]
- effective mass
- topological factor (Bour, Hammer, Lee, Meißner 2013)
Neutron-Deuteron Phase
Shift
0
δ0(degree)
-20
breakup
-1
b=1/100 MeV
-1
b=1/150 MeV
-1
b=1/200 MeV
STM
-40
-60
Pine, Lee, Rupak, EPJA 2013
-80
0
50
p (MeV)
100
Now, what can I do with an adiabatic
Hamiltonian?
Primordial Deuterium
p(n, )d
Warm Up p(n, )d
=
+···
+
Exact analytic continuum result
1
MC (✏) = 2
p +
2
1
(1/a + ip✏ )(
ip✏ )
p
, p✏ = p2 + iM ✏
, MC reduces to known M1 result
Rupak, 2000
Rupak & Lee, PRL 2013
When ✏ ! 0
+
Lattice p(n, )d
Write h
M(✏) =
✓
B |OEM | i i
2
p
M
E
i✏
using retarded Green’s function
◆X
x,y
⇤
B (y)hy|
E
1
ip·x
|xie
ˆ s + i✏
H
cluster Hamiltonian goes here
LSZ reduction in QM
Continuum Extrapolation
0
|M|
1.15
1.1
1.05
0.05
0.1
0.15
s0 = 1.012(18)
s0 = 1.009(16)
s0 = 1.005(07)
p = 15.7 MeV
p = 29.6 MeV
p = 70.2 MeV
1
1.05
= ✏M/p2
1
φ
0.95
0.9
0.85
0.8
0
Fit : s0 + s1
s0 = 1.022(15)
s0 = 1.031(02)
s0 = 1.031(02)
p = 15.7 MeV
p = 29.6 MeV
p = 70.2 MeV
0.1
0.05
δ
0.15
|M| (MeV−2 )
Capture Amplitude
10
10
20
50
70
100
-2
10
10
-4
10
continuum results
-3
δ = 0.6
δ = 0.4
δ=0
δ = 0.6
δ = 0.4
δ=0
-5
= ✏M/p2
φ (deg)
80
60
δ = 0.6
δ = 0.4
δ=0
δ = 0.6
δ = 0.4
δ=0
40
20
0
10
20
50
70
100
p (MeV)
Rupak & Lee, PRL 2013
Something still missing ...
long range Coulomb
···
O
✓
↵
p2
◆
O
✓
↵ ↵µ
p2 p
◆
Proton-Proton Fusion
+
p + p ! d + e + ⌫e
long and steady burning
Proton fusion in
continuum EFT
=
+
+···
Coulomb ladder
Peripheral scattering but how to calculate on
the lattice?
Consider elastic proton-proton scattering as a warmup
Coulomb Subtracted
Phase Shift
60
δsc (degree)
50
40
3% error in fits
Analytic
b=1/100 MeV−1
b=1/200 MeV−1
T =Tc + Tsc
30
20
10
0
0
Rupak, Ravi PLB 2014
5
10
15
p (MeV)
20
25
2⇡ e2i
1
Tc ⇡
µ
2ip
2⇡ e2i( + sc )
T ⇡
µ
2ip
30
Hard spherical wall boundary conditions, Borasoy et al. 2007
Carlson et al. 1984
1
Proton-proton fusion
3% fitting error
propagates
3.2
3
2
8⇡C⌘2
|Tf i (p)|
Gamow peak
6 keV
Λ
⇤(p) =
s
3.4
Analytic
2.8
b=1/100 MeV−1
2.6
b=1/200 MeV
−1
2.4
2.2
2
0
200
⇤EF T (0) ⇡2.51
400
600
E (keV)
800
1000
Kong, Ravndal 1999
Lattice fit : ⇤(0) ⇡2.49 ± 0.02 Rupak, Ravi PLB 2014
Dimer-dimer scattering
Zero range interaction
=
+
=
+
=
+
+
Low momentum : p cot =
+
+...
1
1
+ rdd p2 + · · ·
add
2
What are the ere
parameters?
-- Only scale: a↵
-- Universal numbers: add /a↵ , rdd /a↵
Petrov, Salomon, Shlyapnikov 2005
add = 0.6a↵
add /a↵ ⇡ 0.656
+
Rupak 2006
O(ϵ2)
O(ϵ3)
Rest of the ERE are unknown. Not even their sign!
Luscher’s Method Again
1
S(⌘), ⌘ =
p cot =
⇡L
2
Edd
p
=
2µ⇤
2B + ⌧ (p)[2B
B (MeV)
Different a↵
✓
pL
2⇡
◆2
2BL ]
L (2 fm) QMC(MeV) Exact(MeV)
5
5
-9.790(6)
-9.784
5
6
-9.868
3.5
5
-9.857(6)
9
-6.701(9)
3.5
6
-6.846(12)
-6.848
3
5
-5.630(9)
-5.634
3
6
-5.824(12)
-5.827
-6.693
Dimer-Dimer phase shift
0.0
Universal behavior,
positive effective range
-0.5
aff p cot d
-1.0
ÊÏ ‡
ʇ
Ï
-1.5
‡Ï
Ê
χ
Ê
‡
Ï
‡
Ê
Ï
-2.0
Preliminary
Elhatisari, Katterjohn, Lee, Rupak
-2.5
-3.0
0.0
0.5
1.0
Haff pL2
1.5
2.0
Ê
Dimer-Dimer phase shift
a0 = 0.564±0.028
r0 = 4.288±1.062
0
−0.5
−1
p/κ cot(δDD)
−1.5
−2
−2.5
Preliminary
Elhatisari, Katterjohn, Lee, Rupak
0
0.1
0.2
0.3
κ.alatt.
0.4
0.5
0.6
0.7
0.8 0
1
2
3
4
5
6
7
8
p2/κ2
p
 = 2µB
Conclusions
• Adiabatic Projection Method to derive effective
two-body Hamiltonian
• Retarded Green’s function for problems without
long-range Coulomb
• Spherical wall method with adiabatic Hamiltonian
when long range Coulomb important
• Dimer-dimer scattering
Thank you